We study the homogenization of a Hamilton-Jacobi equation forced by rapidly oscillating noise that is colored in space and white in time. It is shown that the homogenized equation is deterministic, and, in general, the noise has an enhancement effect, for which we provide a quantitative estimate. As an application, we consider Hamilton-Jacobi equations forced by a noise term with small amplitude, and, in increasing the strength of the noise, we observe a sharp transition at which the macroscopic enhancement effect is felt. The results depend on new, probabilistic estimates for the large scale Hölder regularity of the solutions of stochastically forced Hamilton-Jacobi equations, which are of independent interest.